A definable continuous rank for nonmultidimensional superstable theories

Pillay studied nonmultidimensional superstable theories in [8], among other things defining a certain hierarchy of regular types in terms of which all other types may be analysed. Using this hierarchy, he showed that after naming a suitable ‘base’ of parameters, there are j-constructible (hence locally atomic) models over arbitrary sets (see Section 2 for definitions). It is asked at the end of [8] whether the parameter set can be removed. On a different note, it has been known for some time that in nonmultidimensional superstable theories, R∞-rank is definable for formulas having finite rank (see for example [9]). Definability of R∞-rank has had various applications in the literature, and so it is natural to ask whether the restriction to finite rank is necessary. In this paper we do not quite answer this question, but instead use Pillay's analysis to establish the existence of a ‘new’ continuous rank (the original idea for which is due to Tanović) which is defined on all complete types, reflects forking as does R∞-rank and satisfies certain definability properties.

Countable models of trivial theories which admit finite coding

We prove:Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding hasnonisomorphic countable models.Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

Some remarks on nonmultidimensional superstable theories

In this paper we study nonmultidimensional superstable theories T, possibly in an uncountable language, and develop some techniques permitting the generalisation of certain results from the finite rank (and/or countable language) context to the general case.We prove, among other things, the following: there is a set A0 of parameters, which has cardinality at most ∣T∣, and in the finite-dimensional case is finite, such that over any B ⊇ A0 there is a locally atomic model. One of the consequences of this is that if C is the monster model of T, φ(x) is a formula over A0, φC ⊇ X and (X, φC) satisfies the Tarski-Vaught condition after adding names for A0, then there is an elementary substructure M of C containing A0 such that φM = X. Applications to the spectrum problem will appear in [Ch-P].In fact, all the components of the machinery we develop are already present in the general theory. One such component involves a stratification of the regular types of T using a generalized notion of weakly minimal formula. This appears in [Sh, Chapter V and the proof of IX.2.4] and also in [P2]. A second component involves definable groups which arise as ‘binding” groups. The existence of such groups, under certain hypotheses on the behavior of nonorthogonality, is due to Hrushovski [Hr1], and our use of them to help obtain “j-constructible” models is similar to their use in [Bu-Sh].